If I understand your question correctly, you make $n$ independent trials, but split these up into $n=n_1+n_2+n_3$ and compute $k_1$, $k_2$, $k_3$. As BruceET already mentioned this is not a good idea, because you use your data in an inefficient way and obtain too large confidence intervals.
I can only guess what your reasoning might be behind splitting up the trials; maybe it is based on biology textbooks which, according to this ComputationlSciene question recommend this approach for error estimation. The idea behind this suggestion is to use the different results for cross-validation, but there are always better methods (see the discussion of the above question). In this particular case, this is pointless, because many good confidence intervals are known for a binomial $p$, see e.g.
L. D. Brown, T. T. Cai, and A. DasGupta, “Interval estimation for a binomial proportion,”Statistical science, vol. 16, no. 2, pp. 101–117, 2001
Brown et al. recommend the Wilson interval, but if $p$ is close to one or zero, the Bayesian HPD interval has even better coverage probability. It can only be computed numerically, though. So if you need a closed formualy, go for the Wilson interval.